The thresholds for this pet are currently at level 99, virtual level 120, and 200,000,000 experience.
You will not obtain any other skilling pets aside the Dungeoneering skill pet whilst training the Dungeoneering skill.
Skilling pets CAN be obtained whilst training on portables, using skillchompas, or the Crystallise Seren spell.
You CANNOT get any skill pet while using protean items, playing Distractions and Diversions, playing mini-games, by using Experience lamps and stars, or Silverhawk boots.
The Voice of Seren does NOT increase your chances at pets while training in Prifddinas, however you can still obtain the pet while skilling there.